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Register a new userThe Cotangent Bundle of a Cominuscule Grassmannian
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A theorem of the first author states that the cotangent bundle of the type $A$ Grassmannian variety can be embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety. We extend this result to cotangent bundles of cominuscule generalized Grassmannians of arbitrary Lie type.
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We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.
In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincar{e}-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincar{e}-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.
Let X be an irreducible smooth projective curve, of genus at least two, over an algebraically closed field k. Let $mathcal{M}^d_G$ denote the moduli stack of principal G-bundles over X of fixed topological type $d in pi_1(G)$, where G is any almost simple affine algebraic group over k. We prove that the universal bundle over $X times mathcal{M}^d_G$ is stable with respect to any polarization on $X times mathcal{M}^d_G$. A similar result is proved for the Poincare adjoint bundle over $X times M_G^{d, rs}$, where $M_G^{d, rs}$ is the coarse moduli space of regularly stable principal G-bundles over X of fixed topological type d.
In this paper, we consider a generalization of the theory of Higgs bundles over a smooth complex projective curve in which the twisting of the Higgs field by the canonical bundle of the curve is replaced by a rank 2 vector bundle. We define a Hitchin map and give a spectral correspondence. We also state a Hitchin-Kobayashi correspondence for a generalization of the Hitchin equations to this situation. In a certain sense, this theory lies halfway between the theories of Higgs bundles on a curve and on a higher dimensional variety.
We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagatas criterion, localizing the coordinate ring at a suitable set of Plucker coordinates. We prove that these Plucker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.
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